Series And Parallel Circuits

Series and '''parallel''' Electrical Circuit s are two basic ways of wiring components. The naming describes the method of attaching components, i.e. one after the other, or next to each other. It is said that two circuit elements are connected in ''parallel'' if the ends of one circuit element are connected directly (i.e. a conductor) to the corresponding ends of the other. However, when the circuit elements are connected end to end, it is said that they are connected in ''series''. As a demonstration, consider a very simple circuit consisting of two lightbulbs and one 9 V Battery . If a wire joins the battery to one bulb, to the next bulb, then back to the battery, in one continuous loop, the bulbs are said to be in series. If, on the other hand, each bulb is wired separately to the battery in two loops, the bulbs are said to be in parallel. The measurable quantities used here are ''R'', Resistance , measured in Ohms (Ω), ''I'', current, measured in Ampere s (A) ( Coulomb s per Second ), and ''V'', voltage, measured in Volt s (V) ( Joule s per Coulomb ). SERIES CIRCUITS Series circuits are sometimes called ''cascade''-coupled or Daisy Chain -coupled. The Current that enters a series circuit has to flow through every element in the circuit. Therefore, all elements in a series connection have equal currents. Two Ammeter s placed anywhere in the circuit would prove this. Resistors To find the total Resistance of all the components, add together the individual resistances of each component: : : $R_\mathrm{total} = R_1 + R_2 + \cdots + R_n$ :for components in series, having resistances $\ R_1$ , $\ R_2$ , etc. To find the current, $\ I$ use Ohm's Law $I = rac{V}{R_{total}}$ To find the Voltage across any particular component with resistance $\ R_i$ , use Ohm's law again. $V_i = I \cdot R_i$ :Where $\ I$ is the current, as calculated above. Note that the components divide the voltage according to their resistances, so, in the case of two resistors: : $rac{V_1}{V_2} = rac{R_1}{R_2}$ Inductors Inductor s follow the same law, in that the total Inductance of non-coupled inductors in series is equal to the sum of their individual inductances: : : $L_\mathrm{total} = L_1 + L_2 + \cdots + L_n$ However, in some situations it is difficult to prevent adjacent inductors from influencing each other, as the magnetic field of one device couples with the windings of its neighbours. This influence is defined by the mutual inductance M. For example, if you have two inductors in series, there are two possible equivalent inductances: : $\ L_\mathrm{total} = (L_1 + M) + (L_2 + M)$ or : $\ L_\mathrm{total} = (L_1 - M) + (L_2 - M)$ Which formula is the correct one, depends how the magnetic fields of both inductors influence each other. When there are more than two inductors, it gets more complicated, since you have to take into account the mutual inductance of each of them and how each coils influences the other. So for three coils, there are three mutual inductances ( $M_{12}, M_{13}$ and $M_{23}$ ) and eight possible equations. Capacitors Capacitor s follow a different law. The total Capacitance of capacitors in series is equal to the Reciprocal of the sum of the reciprocals of their individual capacitances: : : ${1\over{C_\mathrm{total}}} = {1\over{C_1}} + {1\over{C_2}} + \cdots + {1\over{C_n}}$ The working voltage of a series combination of identical capacitors is equal to the sum of voltage ratings of individual capacitors provided that equalizing resistors are used to ensure equal voltage division. PARALLEL CIRCUITS Voltages across components in parallel with each other are the same in magnitude and they also have identical polarities. Hence, the same voltage variable is used for all circuits elements in such a circuit. To find the total current, ''I'', use Ohm's Law on each loop, then sum. (See Kirchhoff's Circuit Laws for an explanation of why this works). Factoring out the voltage (which, again, is the same across parallel components) gives: : $I_\mathrm{total} = V \cdot \left(rac{1} {R_1} + rac{1} {R_2} + \cdots + rac{1} {R_n} ight)$ Notation
	:<math> R \mathrm{total}	R_1 \ R_2 = {R_1 R_2 \over R_1 + R_2} </math>

:for components in parallel, having resistances ''R''₁, ''R''₂, etc.

''The above rule can be calculated by using Ohm's law for the whole circuit''

:

R_\mathrm{total} = V / I_\mathrm{total}

''and substituting for'' ''I''_total

To find the Current in any particular component with resistance ''R''_i, use Ohm's law again.

:

I_i = V / R_i

Note, that the components divide the current according to their ''reciprocal'' resistances, so, in the case of two resistors:

:

I_1 / I_2 = R_2 / R_1

Inductors

Inductor s follow the same law, in that the total Inductance of non-coupled inductors in parallel is equal to the reciprocal of the sum of the reciprocals of their individual inductances:

:

:

{1\over{L_\mathrm{total}}} = {1\over{L_1}} + {1\over{L_2}} + \cdots + {1\over{L_n}}

Once again, if the inductors are situated in each others' magnetic fields, one has to take into account mutual inductance. If the mutual inductance between two coils in parallel is M then the equivalent inductor is:

:

{1 \over L_\mathrm{total}} = {1 \over (L_1 + M)} + {1 \over (L_2 + M)}

or
:

{1 \over L_\mathrm{total}} = {1 \over (L_1 - M)} + {1 \over (L_2 - M)}

And once again, which formula is the correct one, depends how the magnetic fields of both inductors influence each other.

The principle is the same for more than two inductors, but you now have to take into account the mutual inductance of each inductor on each other inductor and how they influence each other. So for three coils, there are three mutual inductances (

M_{12}, M_{13}

and

M_{23}

) and eight possible equations.

Capacitors

Capacitor s follow a different law. The total Capacitance of capacitors in parallel is equal to the sum of their individual capacitances:

:

:

C_\mathrm{total} = C_1 + C_2 + \cdots + C_n

The working voltage of a parallel combination of capacitors is always limited by the smallest working voltage of an individual capacitor.

SEE ALSO

Y-Δ Transform (a.k.a. ''Star-Triangle transformation'')

Voltage Divider

Current Divider

Electrical Impedance#Combining Impedances

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Series And Parallel Circuits